Can someone explain to me how I can save the colors I made in the PART file in the drawing as well?

Hi! You can't directly export colors from part to drawing, because all lines will be export on one layer (you haven't got layers in part). But you can:
1) create layers with its own colours in drawing
2) try to export part in dwg/dxf (if you have 2022 SW colours will be saved)

For saving colours while printing: File-Save as-PDF, then check print settings (see atachment)